deimurv - neural virtual sensor for the inferential prediction of product quality...

Neural virtual sensor for the inferential prediction of product qualityfrom process variables

R. Rallo a, J. Ferre-Giné a, A. Arenas a, Francesc Giralt b,*a Departament d’Enginyeria Informàtica i Matemàtiques, Escola Tècnica Superior d’Enginyeria, Universitat Rovira i Virgili, Avinguda dels Paı̈sos

Catalans, 26, Campus Sescelades, 43007 Tarragona, Catalunya, Spainb Departament d’Enginyeria Quı́mica, Escola Tècnica Superior d’Enginyeria Quı́mica (ETSEQ), Universitat Rovira i Virgili, Avinguda dels Paı̈sos

Catalans, 26, Campus Sescelades, 43007 Tarragona, Catalunya, Spain

Received 20 April 2000; received in revised form 5 July 2002; accepted 5 July 2002

Abstract

A predictive Fuzzy ARTMAP neural system and two hybrid networks, each combining a dynamic unsupervised classifier with a

different kind of supervised mechanism, were applied to develop virtual sensor systems capable of inferring the properties of

manufactured products from real process variables. A new method to construct dynamically the unsupervised layer was developed.

A sensitivity analysis was carried out by means of self-organizing maps to select the most relevant process features and to reduce the

number of input variables into the model. The prediction of the melt index (MI) or quality of six different LDPE grades produced in

a tubular reactor was taken as a case study. The MI inferred from the most relevant process variables measured at the beginning of

the process cycle deviated 5% from on-line MI values for single grade neural sensors and 7% for composite neural models valid for

all grades simultaneously.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Inferential prediction; Virtual sensors; Selection of variables; Radial basis function; Kohonen neural networks; Fuzzy ARTMAP;

Polyethylene

1. Introduction

Neural network systems have been widely used to

model and control dynamic processes because of their

extremely powerful adaptive capabilities in response to

nonlinear behaviors (Barto, Sutton, & Anderson, 1983).

This adaptability is the result of the architectures

themselves (parallel and/or with classification capabil-

ities) and of the learning algorithms used, which are

biologically inspired and incorporate some aspects of

the organization and functionality of brain neuron cells

(Hertz, Krogh, & Palmer, 1991). As a result, neural

systems can mimic high-level cognitive tasks present in

human behavior, and operate in ill-defined and time-

varying environments with a minimum amount of

human intervention. For example, they are capable of

(i) learning from the interaction with the environment,

without restrictions to capture any kind of functional

relationship between information patterns if enough

training information is provided, (ii) generalizing the

learned information to similar situations never seen

before, and (iii) possessing a good degree of fault

tolerance, mainly due to their intrinsic massive parallel

layout. These properties make neural computing appeal-

ing to many fields of engineering.

The application of neural systems is especially inter-

esting to control and to optimize chemical plants (Hunt,

Sbarbaro, Zbikowski, & Gawthrop, 1992) since the kind

of time-dependent problems dealt with in process

engineering are highly non-linear and, thus, it is difficult

to obtain detailed predictions from first principle models

in real time. A specific area of intrinsic interest to

chemical manufacturing processes is the prediction of

the quality of final products. This is even more vital in

cases where it is difficult to implement reliable and fast

on-line analyzers to measure relevant product properties

* Corresponding author. Tel.: �/34-977-559-638; fax: �/34-977-558-205

E-mail address: [email protected] (F. Giralt).

Computers and Chemical Engineering 26 (2002) 1735�/1754

www.elsevier.com/locate/compchemeng

0098-1354/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 8 - 1 3 5 4 ( 0 2 ) 0 0 1 4 8 - 5

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and to establish appropriate control strategies for

production. Such situations can lead to a significant

production of off-grades, especially during the on-line

operations involved to change product specifications.An alternative is to develop on-line estimators of

product quality based on available process information.

One of the most powerful and increasingly used

methodologies is the inferential measurement (Martin,

1997). This method consists in the forecast of product

quality or of difficult to measure process indicators from

other more reliable or easily performed plant measure-

ments, such as pressures, flow rates, concentrations ortemperatures.

The purpose of the current study is to develop a

virtual sensor to infer product quality from other more

easily measured process variables using several adaptive

neural network architectures as well as different techni-

ques for data preprocessing, including the selection of

variables and the construction of appropriate training

and test sets. The networks that have been consideredare a predictive Fuzzy ARTMAP neural classifier, and a

hybrid network that combines a new strategy to con-

struct dynamic unsupervised classifiers with a supervised

predictor. The neural virtual sensors (soft-sensors)

developed have been applied to infer the Melt Index

(MI) of different low-density polyethylene (LDPE)

grades measured on-line in operating plants. The paper

is organized as follows. In Section 2 all aspectsconcerning the design and implementation of virtual

sensors systems are described, highlighting several

techniques for data preprocessing, their basic architec-

tures and proposed modifications, learning algorithms

and major drawbacks. The performance of sensors has

been illustrated and evaluated with a case study of

LDPE quality inference. The results obtained are then

discussed and concluding remarks about the design andimplementation of virtual sensors systems are finally

presented.

2. Neural virtual sensor

A virtual sensor is a conceptual device whose output

or inferred variable can be modeled in terms of other

parameters relevant to the same process. The software(sensor) should be conceived at the highest cognitive

level of abstraction so that a sufficiently accurate

characterization of the total system behavior could be

attained in terms of errors between the validated or

measured data and the predicted outputs. Artificial

neural networks are an adequate choice because, in

addition to the above, they can improve performance

with time, i.e. are capable of learning real cause�/effectrelations between sensor’s stimulus and its response

when historical databases of the whole process are used

for training. Fig. 1 shows a generic virtual sensor

implementation within a manufacturing process. It can

receive real-time readings of several process variables as

well as feedback signals of downstream on-line analyzers

for the target property. Both sets of data are needed fortraining the virtual sensor. Once trained, this virtual

device uses only real time measurements of the selected

process variables obtained by process sensors at certain

times to infer the value of the product target property.

The output can be redirected as information to the plant

operator or to the control system to maintain optimal

plant operation for a given product quality.

In the current study the aim is the design of a genericmodel for a virtual sensor able to predict the quality of a

product based in the state of the process plant when the

production cycle begins. The virtual sensor thus acts as

an inferential measurement system that anticipates the

downstream target variable as a function of other

process variables measured when production starts.

The dynamic characteristics of chemical plants and the

level of accuracy for the output usually impose certainrequirements to the neural system: (i) adaptive in

architecture so that more processing units could be

added during the training process when needed; (ii)

robust under uncertainty of the input data due to noise

or product grade transitions; and (iii) re-trainable if a

new mode of operation never seen before occurs.

3. Data preprocessing

The inferential measurement systems based on neural

networks are mostly developed using ‘‘data-based’’

methodologies, i.e. the models used to infer the value

of target variables are developed using ‘live’ plant data

measured from the process plant. This implies that

inferential systems are heavily influenced by the quality

of the data used to develop their internal models.Consequently, the first step to build an inferential

measurement system is the preprocessing of data.

During this preprocessing stage two kinds of actions

are performed, first data cleaning and conditioning, and

second the selection of the most relevant information to

develop the model.

It is important to consider that usually the data

acquired from field sensors are noisy and containerroneous or spurious values. To condition these data

all the faulty values must be discarded using filtering

techniques like low-pass filters. Moreover, as the values

are heterogeneous, they must be scaled into a range

suitable for data processing. In addition, in complex

industrial processes, such as chemical processing plants,

the number of plant variables that can be measured is

very large and the sampling rates used for thesemeasurements are high. This implies the generation of

large datasets containing a large amount of features. In

those situations it is very useful to have an ‘‘intelligent

R. Rallo et al. / Computers and Chemical Engineering 26 (2002) 1735�/17541736

system’’ capable of selecting the most relevant features

and examples among all the available information with

the purpose of optimizing the resources needed to build

an accurate and reliable model for the process under

consideration.In the process of building a neural inferential mea-

surement system a reduction in the dimension of the

input space would simplify the input layer of the neural

architecture and reduce the time needed for training.

Additionally, if the class to which a certain input pattern

belongs is known it should be possible to figure out the

features that best discriminate between the different

values of the target properties. The present study adaptsthe technique proposed by Espinosa, Arenas, and Giralt

(2001b) to perform the selection of relevant variables to

the case where the target variable (MI) fluctuates

around a mean value. The discrimination of the training

datasets from the complete set of plant measurements

has been carried out following the work of Espinosa,

Yaffe, Arenas, Cohen, and Giralt (2001a) to include all

relevant information.

3.1. Selection of variables

A sensitivity analysis to reduce the dimension of the

input space could be performed by using either projec-tion techniques, which imply the definition on new

combined variables, or by reducing the number of input

variables while preserving the most relevant features of

the complete input space. The first approach typically

uses statistical analysis (descriptive statistics, cross-

correlations, factorial analysis, PCA, etc.) to find

relations among all variables and to select new repre-

sentative prototypes from the subsets of related variablecombinations. This process entails the projection of the

input space into a lower dimension output space without

the loss of significant information. The main drawback

of this approach is that the newly created variables are

difficult to interpret in terms of the primitive process

variables. Also, orthogonal decomposition may not be

the best approach to reduce the dimension of the input

space, as has been found in pattern recognition pro-

blems related to structure identification in turbulent

flows (Ferre and Giralt, 1993).

The method proposed here projects all subsets of the

input space, with process variables (vi) ordered accord-

ing to relevance and each subset complemented with the

target variable (P ), onto the space generated by a self-

organizing map (SOM) (Kohonen, 1990), which is a

topology-preserving clustering process. The comparison

of each map with the rest maps by means of a

dissimilarity measure (see Appendix A and Kaski &

Lagus, 1997) informs about the relevance of each

combination of input variables in relation to the target

variable. The key of this selection process resides in the

similarity measure used and in the strategy to build the

subsets of variables from the original input space.

The first step in the process of best feature selection

was to define a strategy to populate the set of ordered

variables, S from the set of all N input variables, V�/{v1, v2, . . ., vN}. This procedure can be formalized eitherfollowing the selection (ordering) criterion proposed by

Espinosa et al. (2001b) to develop Quantitative Struc-

ture Property/Activity Relationships (QSPR/QSAR)

models or by the (absolute) value of the correlation of

each input variable with the target or quality property,

½corr(vi , P )½. The latter has been adopted here since thecurrent problem is simpler and does not require an exact

classification of plant events according to a prefixed

knowledge, as it is the case in QSPR/QSAR models

where similar chemicals should be classified into similar

classes. Also, P values and plant operating conditions

should oscillate around their prefixed mean values in a

properly operating chemical plant and correlation

Fig. 1. Sketch of a generic virtual sensor implementation.

R. Rallo et al. / Computers and Chemical Engineering 26 (2002) 1735�/1754 1737

information between these fluctuating signals alone

should be sufficient to order input variables. Using

this correlation approach all variables are ranked by

correlation and the input set V were reordered so that,S�/{s1, . . ., sN}, where N is the cardinality of S andcorr(si , P )]/corr(si�1, P ) �/ i : 1, . . ., N�/1.

Let us now consider the selection of the most

adequate subset of input variables from S , i.e. the

minimum set containing all relevant information. SOMs

were determined for all N subsets that can be generated

from S by successively adding to the first subset formed

by the first element and the target variable, S1�/{s1, P}the rest of ordered variables, until the last subset SN �/{s1, . . ., sN , P} was formed. The corresponding SOMsfor all Si �/ i : 1, . . ., N�/1 were trained and thedissimilarity D(L,M) between every pair of maps

(L,M) was computed using Eq. (A3) in Appendix A.

The average dissimilarity with all the rest of the possible

variable configurations was computed and the Si yield-

ing the minimum average dissimilarity was selected asthe set of minimum dimension containing the most

relevant features of the input space This selected set can

be considered as the subset of variables most similar, in

terms of information contents about the target property,

to all other possible ordered subsets formed from the

input variables.

3.2. Training set

As some variables are more relevant than others, also

some input patterns of data may be more unique than

others and should be considered for training. The

appropriate selection of patterns for the training is one

of the most important tasks in machine learning.

Different strategies can be used for the selection of the

most suitable training set of data among all the available

process information. One method to construct thetraining set consists in the selection of data from the

time series of recorded plant data while keeping the

remaining data sequence to test the performance of the

sensor. This facilitates the representation of data and the

evaluation as if the sensor was operating under real

plant conditions, predicting the target property sequen-

tially in time. This selection procedure does not assure,

however, that all significant, singular or redundant,information was used for sensor development during

training.

An alternative is to select the most suitable training

and test sets from the complete pool of available

patterns contained in the time-records, independently

of their position in the time-sequences. This assures that

all significant, singular or redundant, information is

presented to the sensor during the learning stage andthat testing is performed with data that, while not

previously seen by the sensor, has similar characteristics

to those belonging to the classes or categories used for

training. This requires the application of pre-classifica-

tion algorithms such as fuzzy ART, as suggested by

Espinosa et al. (2001a), or the dynamic clustering

procedure outlined in Section 4. Patterns located inregions of the process state space with low ‘‘population

density’’ are labeled as candidates for training and those

located in high density regions as candidates for testing.

Test sets are finally built by randomly selecting a certain

number of patterns labeled as test candidates.

The pre-screening option chosen in the present study

is the application of the algorithmic method proposed

by Tambourini and Davoli (1994) because with smalltraining sets it assures good network generalization

capabilities. The key idea is to include as learning

examples those that exhibit classification difficulties

during training. The hypothesis made is that these

patterns contain most of the important classification

information about the problem under study. They

presumably are the patterns that best represent the

different problem classes and have to be included in thetraining set to supplying the network with good

examples. If the network learns successfully these

training set patterns, it could be expected to perform

well with unknown patterns, since it has already learned

most of the characteristics that distinguish the problem

classes.

4. Neural architectures

Three neural models have been developed and eval-

uated to build the virtual sensor. One is based on a

predictive Fuzzy ARTMAP architecture that has been

capable of learning the dynamics of large-scale struc-

tures in a turbulent flow (Giralt, Arenas, Ferre-Giné,

Rallo, & Kopp, 2000) and to develop successful QSPR

and QSAR models for the prediction of physicochemicalproperties and biological activities (Espinosa et al.,

2001a; Espinosa et al., 2001b). The other two sensors

are based on two supervised algorithms that connect a

Radial Basis Function (RBF) layer with the required

output. The RBF layer has been constructed following a

new procedure that allows dynamic network growing

during the training stage. Outputs are either the average

target property value for all the input patterns belongingto the cluster that is activated or the target property

value that results from the incorporation of RBFs at the

cluster centers.

4.1. Fuzzy ARTMAP

The Fuzzy ARTMAP neural network is formed by a

pair of fuzzy ART modules, Art_a and Art_b, linked byan associative memory and an internal controller

(Carpenter, Grossberg, Marcuzon, Reinolds, & Rosen,

1992), as shown in Fig. 2. The Fuzzy ART architecture


was designed by Carpenter, Grossberg, and Rosen

(1991) as a classifier for multidimensional data cluster-

ing based on a set of features. The elements of the set ofn -dimensional data vectors {j1, . . ., jp}, where p is thenumber of vectors to be classified, must be interpreted as

a pattern of values showing the extent to which each

feature is present. Every pattern must be normalized to

satisfy the following conditions:

ji � [0; 1]n

Xnj�1

jij �k � i�1; . . . ; p(1)

The classification procedure of fuzzy ART is based on

Fuzzy Set Theory (Zadeh, 1965).

The similarity between two vectors can be established

by the grade of the membership function, which for two

sets (l , m ) of generic vectors can be easily calculated as

grade(jl ƒjm)�jjl ffl jmj

jjlj(2)

In Eq. (3) the fuzzy AND operator ffl/ is defined by,

ffl:[0; 1]n� [0; 1]n 0 [0; 1]n

(jl ; jm) 0 ji(3)

The components of the image vector that results from

this application (3) are

jij �min(jlj; j

mj ) � j�1; . . . ; n (4)

The norm ½ �/½ in Eq. (3) is the sum of the components ofthe vector given by Eq. (4).

The classification algorithm clusters the data into

groups or classes with a value for the grade of member-

ship in Eq. (2) greater than the vigilance parameter r .

The value of r controls the granularity of the classes

and allows the implementation of a desired accuracy

criterion in the classification procedure. A weight vectorvm represents each class m . The procedure starts by

creating the first class from the first pattern presented to

the network,

v1�j1 (5)

The rest of input patterns ji (i�/2, . . ., p) are thenpresented to the network and if the similarity of ji with

any established class m is greater than r then ji is

classified into this class, and the representative of this

class is updated according to

vmnew�vmoldfflj

i (6)

Otherwise a new class represented by ji is created. Eq.(6) is the learning rule of the net. The mechanisms to

speed up the process and to conduct the classification

properly can be found elsewhere (Carpenter et al., 1991).

The dynamics of Fuzzy ARTMAP are essentially the

same as two separate Fuzzy ART networks, each one

working with a part of the training information. The

first part could be interpreted as the vector input pattern

and the second one as the desired classification output(supervisor). The associative memory records the link

between the classes corresponding to the input pattern

and the desired classification. The internal controller

supervises if a new link is in contradiction with any

other previously recorded. If no contradiction is found,

the link is recorded; otherwise the pattern is re-classified

with a larger vigilance parameter. Once the network has

been trained it can be used to classify input vectorswithout any additional information.

The Fuzzy ARTMAP architecture, which has been

successfully applied to educe the different classes of

large-scale events present in free turbulence (Ferre-Giné,

Rallo, Arenas, & Giralt, 1996), was designed to classify

data and, thus, cannot generate an output pattern after

the training stage. To implement predictive capabilities,

the categories educed by the system from the learnedinformation are linked to the desired outputs, as

depicted in Fig. 2. This is mathematically equivalent to

defining an application from the space of categories to

that of output patterns, the image of the application

being defined by examples of patterns provided to the

neural system in a supervised manner. The accuracy of

the procedure increases asymptotically towards a con-

stant value with the number of examples used fortraining, i.e. when the space of outputs is accurately

mapped. In the predictive mode, only the category layer

of Art_b in Fig. 2 is active and linked to Art _a to

provide an output for each input vector presented to this

module (Giralt et al., 2000).

4.2. Dynamic unsupervised layers

The success of predictive fuzzy ARTMAP in difficult

forecasting problems, together with its limitations when

interpreting information with underlying periodicity(Carpenter et al., 1992) and limited training, has

motivated the search for other potentially suitable

systems for sensor development, such as dynamic

Fig. 2. Modified fuzzy ARTMAP neural network architecture.


unsupervised layers. One of the challenges related with

the application of this neural system is the design of a

node generation mechanism and of a clustering process

that are simple enough and simultaneously compatiblewith different supervised output generation procedures.

4.2.1. Node generation

The current approach to construct the unsupervised

RBF layer is performed in four steps: (i) set-up an initial

configuration with the center of the first cluster formed

by one pattern chosen randomly from the training

dataset; (ii) determine the minimal mean distancebetween patterns; (iii) use this distance as the constant

maximum attention radius dmax to control the genera-

tion of new nodes over the node-generation process; and

(iv) present a new input pattern, j , to the network and

compute the Euclidean distance between the pattern and

all nodes, and adapt the structure of the network

according to the following two rules. If the input pattern

is located inside the region of influence of any node i ,the pattern is classified and the center of this node is

adapted using a winner takes-all approach based on

Kohonen’s learning rule (a similar approach using k -

means was used by Hwang and Bang (1997) to speed up

the training of RBF layers),

ci(n�1)�ci(n)�a(n)[j�ci(n)] (7)

with n denoting the training epoch, ci(n ) the center

vector of the selected node, and a (n ) a monotonicallydecreasing learning rate computed as a (n )�/a01/�n ,which controls the adaptation or upgrading of the center

of the cluster. Otherwise, if the input pattern is located

outside the region of influence of all the nodes, a new

node is created with the center located at the point that

defines the input pattern. The procedure is repeated

until the number of nodes stabilizes and either the

classification or the number of iterations reaches apredetermined minimum or maximum value, respec-

tively.

The current algorithm tends to create an appropriate

number of clusters since it determines the attention

radius based on the distribution of the training patterns.

This allows the construction of the minimal clustering

necessary to achieve a good classification in accordance

with the attention radius chosen. The process ofcomplying with a given classification accuracy has to

be tested for generalization by trial and error, which is a

customary practice when working with neural systems.

4.2.2. Output generation

Once the classifier is built it is necessary to effect an

output from the unsupervised layer. This procedure

could consist in a hybrid approach that combines thecurrent dynamic unsupervised classifier with a super-

vised learning engine. In this work two techniques for

producing this output are presented. The first is a

clustering average based on the labeling of the unsuper-

vised layer using the values of the target variable. One of

the most common labeling processes consists in aver-

aging the target value for each of the training patternsbelonging to a given cluster, like in the k -means

algorithm. This averaged value is subsequently the

output of the network. This labeling algorithm can be

summarized as follows: (i) Obtain a pattern from the

training set; (ii) compute the winner node; (iii) add its

value to the output value of the winner node; (iv)

increase the pattern counter for this node; (v) repeat (i)

until all patterns have been processed; (vi) compute theaverage output value for each node. Once the dynamic

unsupervised layer is labeled the network is ready to

infer the target property values, i.e. act as a virtual

sensor.

The second technique used in the current study to

obtain an output from the unsupervised layer consists in

the placement of RBF over the cluster centers with

supervised training to adjust the output. This neuralnetwork is herein after identified as Dynamic Radial

Basis Function network (DYNARBF). RBF neural

networks allow the parametrization of any function

f (x ) as a linear combination of non-linear basis func-

tions (Powell, 1987, 1992; Broomhead & Lowe, 1988;

Lee & Kil, 1988; Moody & Darken, 1989; Poggio &

Girosi, 1990a,b; Musavi, Ahmed, Chan, Faris, &

Hummels, 1992),

f (x)�p�X

qjG(kx�xjk) (8)

where j is the function index, the norm ½ �/½ is theEuclidean distance (Park & Sandberg, 1991), xj are the

centers of the proposed basis functions, p and q are

adjustable parameters and G is a radial kernel function.

In the current model a Gaussian activation function is

used,

G(rj)�exp(�r2j =2sj) (9)

where rj is the Euclidean distance to the center of the j-

class and sj is the dispersion of the Gaussian. For each

activation function (9) the center position xj and its

width sj must be determined to define a receptive field

around the node. Both can be determined by an

unsupervised learning process, in which data are clus-tered into ‘‘classes’’ or nodes. The idea is to pave the

input space (or the part of it where the input vectors lie)

with the receptive field of these nodes (Gaussians in this

case).

A map between the RBFs outputs and the desired

process outputs is then constructed in a second super-

vised training stage. It should be noted that the RBF

neural network needs some a-priori hypothesis concern-ing the number of nodes that will be used for any

particular problem. This is a major drawback in the

application of RBFs because the approximation error is


highly dependent on the number of nodes. Usually,

more nodes imply more accuracy in the mapping of the

predicted target values over the training dataset. Never-

theless, ‘‘over-fitting’’ during training could in somecases imply a loss of network generalization capabilities

during testing. There is no a priori methodology to

estimate rigorously the number of nodes for optimal

generalization. These issues have been the subjects of

research in the past (Platt, 1991; Fritzke, 1994a,b;

Berthold & Diamond, 1995). These studies share the

common strategy of extending on-line the structure of

the neural network to reduce a given measure of theclassification error. In this study the Growing Unsu-

pervised Layer explained above is used to overcome this

drawback.

4.2.3. Training procedure

The neural system is trained in two separate phases.

First, the unsupervised layer that defines the number of

radial functions in the hidden layer as well as their

position in input space is constructed. The width of each

radial function sj is usually calculated ad hoc as the

mean Euclidean distance between the k -nearest neigh-bors of node j ,

sj �s0XKi�1

d(xj; xi) (10)

The center of the i-Gaussian is xi , s0 is a constant forwidth scaling, K is the cardinality of the neighborhood

and d the Euclidean distance.

The activation of the functions once placed over the

hidden layer is accomplished by

fj(j)�exp��d2(j; xj)

2s2j

�

Aj(j)�fj(j)XN

k�1

fk(j)

8>>>>>><>>>>>>:

(11)

where j is the pattern presented to the network, Aj the

activation of node j and N the total number of

gaussians.

In a second supervised learning stage the activation ofthe RBF layer for a given input pattern j is related to

the desired output ũ: First, the output of the network iscomputed as,

ui �XNj�1

wijAj(j) (12)

Then the weights are updated using the common

Delta rule minimization error procedure,

Dwij �b(ui� ũi)Aj(j) (13)

Here, the activation of node j is the value of the

Gaussian function when the input i is presented with a

constant learning rate b .

The combination of Eqs. (11) and (12), jointly with

the information contained at the centers (xi) and in thewidths (si ) of the RBFs, and with the weights connect-

ing the activation of each radial function with the output

node (wi), yields a analytical model relating the target

property with the parameters of the neural network:

P�XNi�1

wiexp(�d2(j; xi)=2s

2i )PN

j�1 exp(�d2(j; xj)=2s2j )

(14)

In this equation N is the number of RBF nodes, d the

Euclidean distance and j is the input vector of the

process variables presented to the network

5. Case study: virtual sensor for melt index in LDPE

process plant

5.1. Problem statement

The polymerization of ethylene to produce LDPE is

usually carried out in tubular reactors 800�/1500 m longand 20�/50 mm in diameter at pressures of 600�/3000atmospheres (Chan, Gloor, & Hamielec, 1993; Lines et

al., 1993). The quality of the polymer produced is

determined essentially by the MI, which is measured

by the flow rate of polymer through a die. The on-linemeasurement of this quantity is difficult and requires

close human intervention because the extrusion die often

fools and blocks. As a result, in most plants the MI is

evaluated off-line with an analytical procedure that

takes between 2 and 4 h to complete in the laboratory,

leaving the process without any real-time quality

indicator during this period. Consequently, a model

for estimating the MI on-line would be very useful bothas an on-line sensor and as a forecasting system. In

addition, it would allow the supervision of the overall

process and to avoid any mismatch of product quality

during product grade transitions. However, a model

derived from first principles capable of continuously

predicting accurate MI values for any LDPE process in

real time is still non-existent. Instead, some production

plants use data based, linear and non-linear correlationmodels to overcome this deficiency.

Fig. 3 summarizes the main characteristics of the

LDPE plants studied. Several sets of real data corre-

sponding to several production cycles are analyzed here.

The MI values used here were determined every 10 min

with on-line sensors that were calibrated by off-line

determinations. The error associated with these on-line

MI measurements is 9/2%. Changes in MI correspondto changes in the physical or chemical characteristics of

the desired product (grade transitions). The six LDPE

product grades cluster into three families according to


their average MI values and polymer densities, as shown

in Fig. 4. Each of these families contains two grades.

Also each cluster has a different population density,

with the highest one corresponding to the lowest value

of MI.

A pool of process information, formed by the 25

process variables (pressures, flow rates, temperatures of

the cooling/heating streams of the reactor, etc.) listed in

alphabetical order in Table 1, has been chosen to

develop the virtual sensors. This table also includes the

correlation of each variable with the MI for the six

grades produced. The characterization and the predic-

tion of the time-variation of MI (last variable listed in

Table 1) has been approached with both the complete

set of 25 variables and the reduced sets for all grades,

following the procedure explained in a previous section.

In the case of composite models applicable to all families

simultaneously, these process variables have been com-

plemented with a normalized label to identify grades and

facilitate product transitions during production cycles.

All the time records of the 25 variables have been

acquired by field measurement instruments, and sent to

the control computer of the plant for their processing.

This computer receives voltage signals that are con-

verted into fix point numeric data within a certain range.

Fig. 3. LDPE plant diagram with typical time scales.

Fig. 4. Families of LDPE identified by the variation of MI with

polymer density.

Table 1

Process variables and correlations with the MI for all the LDPE grades

produced.

Variable name Units ½R ½ with MI

Compressor throughput Tm/h 0.044

Concentration 1 % 0.527




Density g/cm3 0.183

Extruder power consumption A 0.583

Extruder speed rpm 0.056

Flow rate 1 kg/h 0.021




Level % 0.126

MI g/10min 1.000

Pressure Kg/cm2 0.052

Temperature 1 8C 0.305Temperature 2 8C 0.023Temperature 3 8C 0.522Temperature 4 8C 0.115Temperature 5 8C 0.122Temperature 6 8C 0.136Temperature 7 8C 0.428Temperature 8 8C 0.446Temperature 9 8C 0.112Volumetric flow rate 1 l/h 0.324

Volumetric flow rate 2 l/h 0.518


The data presented in the following analysis corre-

spond to time intervals of 10 min. The virtual sensors

anticipate the MI values from process variables mea-

sured when the production cycle begins. The duration ofcycles is determined by the mean residence time inside

the reactor (t:/30 min). This choice is independentlyconfirmed by the spectral analysis of plant data for all

grades, which show an underlying periodicity around 1.1

residence time t units.

5.2. Virtual sensor implementation

Fig. 3 depicts the current sensor implementation

within the LDPE process flow sheet. It can receive

real-time readings of process variables as well as feed-

back signals of downstream on-line analyzers; both sets

of data were used for training (and later adapting) the

virtual sensor. Once trained, this virtual device uses only

real time measurements of the selected process variablesmade by process sensors at any time to infer the value of

the product target property when leaving the reactor.

The output can be redirected as information to the plant

operator or to the control system to maintain optimal

plant operation for a given product quality.

The aim of the current study is to implement a virtual

sensor to predict the quality of LDPE, i.e. the target

variable P�/MI, based in the state of the plant. Thevirtual sensor behaves as a black-box model that relates

the MI of produced LDPE to other process variables

measured when the corresponding production cycle

began. It is important to remark that the current

approach is not a time-series analysis, since it is time

independent. The functionality between the output and

the input variables is of the form MI(t�/t)�/F [v1(t),. . ., vn (t)], where each vi(t) is any of the process variableslisted in Table 1, excluding MI, measured at time t .

Nevertheless, since field measurements are available as a

time series the current neural sensors have also been

developed and operated according to production time-

sequences and cycles.

The three types of neural architectures introduced in

the previous sections have been used to develop the

virtual sensor. In addition a fourth linear model hasbeen used as a reference for evaluation purposes. This

linear model is formulated in terms of normalized

variables as,

MI�a0�a1v1� . . .�aNvN �aN�1vN�1 (15)

In this equation aj �/ j : 0, . . ., N�/1 are adjustablecoefficients and vi �/ i : 1, . . ., N are each of the N�/25normalized process variables listed in Table 1 and

measured simultaneously at the beginning of the pro-

duction cycle. The last variable vN�1 in Eq. (15)correspond to a label identifying the product grades,

which is used only for the composite model and its

normalized value identifies product quality and grade.

The values of the coefficients aj in Eq. (15) are not

reported here for brevity. Results showed that high or

low values of these coefficients did not necessarily

correspond with high or low correlations of variableswith MI.

Two types of virtual sensors were developed for each

neural system considered: single models for every

product grade and composite models for all LDPE

grades jointly. Each kind of model was trained and

tested with datasets defined according to the sequential

and pre-classification procedures described in previous

sections, so that the performance of each sensor couldbe assessed as independently as possible of the limited

number of training patterns available. Furthermore, the

original (complete) and reduced sets of variables have

been used in order to investigate the sensitivity of the

input pattern dimension in sensor’s performance. On the

other hand, the composite sensors, applicable to all

grades simultaneously, should provide a good estimate

of the benefits derived from the classification capabil-ities of all the neural systems considered. Table 2

includes the most relevant information concerning the

training and test sets used in the current study to

develop sensors from the complete input dataset of 25

process variables (upper part of Table 2) and from the

reduced sets selected by the sensitivity analysis using

SOMs (lower part of Table 2). From the point of view of

the MI, the two sets of data used for training and testingeach kind of neural sensor, characterized by the different

input dimension, had similar characteristics, i.e. equiva-

lent average MI values, and similar number of different

and of single MI values in the data records. The

parameters used to develop (train and test) the three

current neural sensor models are summarized in Table 3.

5.3. Preprocessing of variables

The data used for training and testing the virtual

sensor have been acquired from the historical logs

recorded in a real LDPE plant for the 25 process

variables and MI listed in Table 1. These variables are

sufficient to capture the dynamics of the LDPE plant for

the six different quality grades (MI) considered, grouped

into three families of final product. This table also

includes the absolute value of their correlation with MI.Data were filtered to discard abnormal situations and to

improve the quality of the inference system. The input

and output variables were normalized with respect to

their maximum operation values for all the LDPE

grades considered, so that� MI; vi � [0; 1]:/Data from the time records of the process variables

and MI were first separated into training and test sets as

indicated in Table 2. The first selection procedure usedwas aimed at preserving the time-series structure of the

recorded plant data for each type of LDPE analyzed.

This sequential procedure consisted in separating from


the time records of all variables the last 100 patterns for

testing. This facilitated the representation of data and

the evaluation as if the sensors were operating under real

plant conditions, predicting MI sequentially in time. The

pre-classification procedure outlined in the previous

section was secondly applied to generate optimized

training sets. The main characteristics of these training

and test sets are also summarized in Table 2. Finally, themethod described before for the selection of the most

relevant features or input variables has been applied in

order to reduce the size of the training set, thus reducing

the CPU time needed to adapt the virtual sensor. The

sets of ordered features for each model are summarized

in Table 4. Note that the 25 input process variables

ordered in Table 4 by correlation do not include the

product grade label, even for the composite modelsituation, since this information is added only in this

case after the best set of reduced variables is selected

from the ordered complete set.

6. Results and discussion

Table 2 summarizes the characteristics of the datasets

used to build and test the different virtual sensors

Table 2

Characteristics of the datasets used for training and testing the four virtual sensors with the complete and the reduced set of input variables

Product families Grade # Patterns (test/train%) MI avg # Different MI values (%) # Single MI values (%)

Sequential Pre-classification Sequential Pre-classification Sequential Pre-classification

Training set for the complete process input information

I A 3043 0.74 0.74 321 (11) 321 (11) 99 (3) 99 (3)

B 3823 0.73 0.72 284 (7) 283 (7) 82 (2) 81 (2)

II C 640 1.76 1.75 250 (39) 277 (43) 104 (16) 115 (18)

D 957 1.74 1.73 261 (27) 273 (29) 113 (12) 115 (12)

III E 1445 2.00 2.00 371 (26) 383 (27) 150 (10) 152 (11)

F 4295 3.46 3.45 721 (17) 730 (17) 243 (6) 241 (6)

I�II�III A�/F 14 203 1.80 1.80 1389 (10) 1406 (10) 292 (2) 287 (2)

Test set for the complete process input information

I A 100 (3.3) 0.73 0.72 69 68 46 50

B 100 (2.6) 0.70 0.75 70 65 50 43

II C 100 (15.6) 1.68 1.73 76 88 57 78

D 100 (10.4) 1.73 1.72 80 74 62 53

III E 100 (6.9) 1.94 1.99 84 84 68 71

F 100 (2.3) 3.46 3.49 71 82 46 67

I�II�III A�/F 600 (4.2) 1.70 1.43 367 (61) 228 (38) 221 (37) 102 (17)

Training set for the reduced process input information

I A 3043 0.74 0.72 321 (11) 305 (10) 99 (3) 85 (3)

B 3823 0.73 0.72 284 (7) 280 (7) 82 (2) 79 (2)

II C 640 1.76 1.72 250 (39) 266 (41) 104 (16) 113 (18)

D 957 1.74 1.72 261 (27) 258 (27) 113 (12) 104 (11)

III E 1445 2.00 2.00 371 (26) 378 (26) 150 (10) 154 (11)

F 4295 3.46 3.50 721 (17) 558 (13) 243 (6) 111 (2)

I�II�III A�/F 14 203 1.80 1.80 1389 (10) 1358 (9) 292 (2) 284 (2)

Test set for the reduced process input information

I A 100 (3.3) 0.73 0.72 69 52 46 34

B 100 (2.6) 0.70 0.72 70 65 50 43

II C 100 (15.6) 1.68 1.77 76 69 57 47

D 100 (10.4) 1.73 1.72 80 46 62 23

III E 100 (6.9) 1.94 2.00 84 77 68 58

F 100 (2.3) 3.46 3.50 71 63 46 38

I�II�III A�/F 600 (4.2) 1.70 1.25 367 (61) 271 (45) 221 (37) 160 (27)

Table 3

Training parameters for the three neural models

Fuzzy

ARTMAP

Clustering

average

DynaRBF

Single grade model r�0.995 dmax�0.05All variables

Composite model dmax�0.05 a0�0.1(in Eq. (7))

All variables

Single grade model r�0.9995 a0�0.1(in Eq. (7))

b�0.1(in Eq. (13))

Reduced variables

Composite model s0�1.0(in Eq. (10))

Reduced variables


developed for three different families of LDPE identified

in Fig. 3 according to their different MI values. Each of

these families includes two product grades that are

produced under different process conditions (dynamics).

Single models for each product grade as well as

composite ones valid for the ensemble of all LDPE

families have been developed to test and compare the

performance of the different sensor currently under

consideration. The abovementioned data preprocessing

techniques aimed at reducing the number of variables

and at optimizing the composition of the training set,

have been applied in all cases. The results obtained for

all virtual sensors considered are summarized in Tables

5 and 6.

It should be noted that the average characteristics of

the training and test sets listed in Table 2 differ slightly

for single grade models and significantly for composite

ones, as illustrated by the average MI values listed for

each case. Differences are even more noticeable when

the pre-classification technique for data selection is

applied, and between models built with either all input

variables or with the most relevant input features only.

This should be kept in mind when comparing relative

errors for different models, particularly for composite

Table 4

Ordered input process variables according to correlation with MI for all product grades

Grade A Grade B Grade C Grade D Grade E Grade F All grades

Temperature 3 Temperature 3 Temperature 2 Flow rate 1 Level Extruder power Flow rate 4

Density Density Temperature 9 Volumetric flow

rate 2

Temperature 6 Volumetric flow

Rate 1

Flow rate 3

Temperature 9 Concentration 1 Volumetric flow

rate 2

Temperature 2 Flow rate 4 Temperature 9 Extruder power

Concentration 1 Volumetric flow

Rate 1

Extruder power Concentration 3 Temperature 7 Temperature 5 Concentration 1

Temperature 8 Temperature 9 Temperature 1 Temperature 9 Temperature 8 Temperature 4 Temperature 3

Extruder speed Extruder power Concentration 1 Temperature 1 Volumetric flow

rate 1

Compressor

throughput

Volumetric flow

rate 2

Temperature 7 Temperature 8 Extruder speed Volumetric flow

rate 1

Concentration 1 Temperature 7 Temperature 8

Concentration 2 Temperature 7 Temperature 3 Concentration 2 Extruder speed Level Temperature 7

Concentration 3 Concentration 4 Density Extruder speed Volumetric flow

rate 2

Temperature 8 Volumetric flow

rate 1

Volumetric flow

rate 1

Volumetric flow

rate 2

Concentration 3 Flow rate 4 Temperature 1 Temperature 3 Temperature 1

Temperature 4 Extruder speed Flow rate 3 Flow rate 2 Compressor

throughput

Concentration 2 Concentration 2

Compressor

throughput

Flow rate 2 Temperature 6 Density Flow rate 3 Density Density

Extruder power Temperature 1 Concentration 2 Concentration 1 Temperature 9 Flow rate 3 Concentration 3

Level Flow rate 1 Level Concentration 4 Concentration 2 Flow rate 2 Temperature 6

Pressure Flow rate 3 Concentration 4 Pressure Temperature 4 Temperature 6 Level

Flow rate 2 Temperature 5 Flow rate 4 Flow rate 3 Density Concentration 1 Temperature 5

Temperature 6 Level Flow rate 2 Level Extruder power Flow rate 1 Temperature 4

Temperature 1 Compressor

throughput

Flow rate 1 Temperature 8 Concentration 4 Flow rate 4 Temperature 9

Temperature 5 Temperature 2 Temperature 4 Temperature 6 Concentration 3 Concentration 4 Extruder speed

Flow rate 1 Concentration 3 Volumetric flow

rate 1

Temperature 3 Temperature 5 Extruder speed Pressure

Temperature 2 Concentration 2 Pressure Extruder power Temperature 2 Temperature 1 Compressor

throughput

Concentration 4 Temperature 4 Temperature 8 Temperature 7 Flow rate 2 Temperature 2 flowrate2

Volumetric flow

rate 2

Pressure Temperature 7 Temperature 5 Temperature 3 Pressure Temperature2

Flow rate 3 Flow rate 4 Compressor

throughput

Compressor

throughput

Flow rate 1 Concentration 3 Flow rate 1

Flow rate 4 Temperature 6 Temperature 5 Temperature 4 Pressure Volumetric flow

rate 2

Concentration 4

The horizontal lines distinguish the sets of reduced process variables (minimum dissimilarity) for each product grade and composite grades.


models. In this case the average MI values of the sparse

test sets will neither coincide with that of the training set

nor with that of any of the six grades considered

individually, and will differ significantly when both

pre-classification and variable reduction techniques are

applied to develop the composite models. For example,

the average MI values for both the sequential and pre-

classification training sets used to develop composite

models with the reduced set of input variables is equal to

1.80 in Table 2 while that for the corresponding test sets

are 1.7 and 1.25, respectively. Thus, only absolute errors

are reported in Tables 5 and 6 for the composite models.

6.1. Models with the complete set of variables

Table 5 summarizes the performance of all neural

sensor models developed with the complete set of input

process variables after training with both the sequential

and the pre-classified sets of data given in the upper part

of Table 2, which also includes the test sets used for

these models. The absolute and relative mean errors

listed in Table 5 for single grade models and sequential

training (upper part) indicate that all neural sensors,

especially DynaRBF, function better than the linear

model on the overall, with absolute mean errors for the

Table 5

Absolute and relative mean errors, and standard deviations for the test sets predicted by all sensor models built with all process variables after

training with both the sequential and the pre-classified set of patterns

Family Grade Linear model Clustering average Fuzzy ARTMAP DynaRBF

Abs (%) Abs (%) Abs (%) Abs (%)

Std. dev. Std. dev. Std. dev. Std. dev.

Sequential

I A 0.100 0.072 0.069 0.066

(13.7) (9.9) (9.4) (9.0)

0.099 0.066 0.052 0.053

B 0.080 0.063 0.073 0.057

(11.4) (9.0) (10.4) (8.1)

0.063 0.065 0.061 0.050

II C 0.510 0.438 0.460 0.456

(30.3) (26.1) (27.4) (27.1)

0.411 0.378 0.385 0.384

D 0.148 0.190 0.163 0.152

(8.5) (10.9) (9.4) (8.8)

0.126 0.132 0.118 0.106

III E 0.323 0.295 0.302 0.286

(16.6) (15.2) (15.6) (14.7)

0.303 0.281 0.276 0.291

F 0.393 0.383 0.358 0.339

(11.3) (11.1) (10.3) (9.8)

0.272 0.262 0.260 0.234

I�II�III A�/F 0.295 0.157 0.163 0.1510.301 0.160 0.152 0.148

Pre-classified

I A 0.107 0.057 0.049 0.048

(14.9) (7.9) (6.8) (6.7)

0.165 0.050 0.045 0.067

B 0.113 0.051 0.038 0.041

(15.1) (6.8) (5.1) (5.5)

0.243 0.053 0.027 0.065

II C 0.195 0.147 0.131 0.149

(11.3) (8.5) (7.6) (8.6)

0.202 0.113 0.116 0.166

D 0.132 0.063 0.058 0.059

(7.8) (3.7) (3.4) (3.4)

0.159 0.065 0.060 0.064

III E 0.164 0.064 0.076 0.056

(8.2) (3.2) (3.8) (2.8)

0.165 0.066 0.077 0.049

F 0.213 0.149 0.165 0.146

(6.1) (4.3) (4.7) (4.2)

0.165 0.151 0.174 0.151

I�II�III A�/F 0.182 0.121 0.118 0.1250.178 0.114 0.120 0.115


linear model ranging from 0.080 to 0.510, respectively,

for I(B) and II(C), and those for the neural sensors from

0.057 to 0.460. The standard deviations listed also in

Table 5 are of the same order of magnitude as the

absolute mean errors for all models with sequential

training.

The better performance of neural sensors is clearer in

family I, where all models yield consistently good

predictions. Increases of approximately 40 and 20% in

prediction accuracy are obtained for grades I(A) and

I(B), respectively, with the neural models compared to

the linear model. Deficient training is the cause for the

anomalous high errors of grades II(C) and III(E) and

for the slightly better performance of the linear model

for grade II(D) since they disappear when the pre-

classified set of data was used for training, as shown in

the lower part of Table 5. Note that the small number of

640 training patterns available for training grade II(C)

(see Table 2) aggravates this situation. In addition, the

training set has the largest test to train ratio (15.6%) and

is the most heterogeneous one, as indicated by its high

percentage of different and single MI values in Table 2.

The fact that an increase of 50% in the number of

training patterns between II(C) and II(D) reduces errors

Table 6

Absolute and relative mean errors, and standard deviations for the test sets predicted by all sensor models built with the reduced set of process

variables after training with both the sequential and the pre-classified set of patterns

Family Grade Linear model Clustering average Fuzzy ARTMAP DynaRBF

Abs (%) Abs (%) Abs (%) Abs (%)

Std. dev. Std. dev. Std. dev. Std. dev.

Sequential

I A 0.083 0.064 0.066 0.059

(11.4) (8.8) (9.0) (8.1)

0.066 0.051 0.061 0.055

B 0.070 0.058 0.070 0.057

(10.0) (8.3) (10.0) (8.1)

0.055 0.052 0.057 0.049

II C 0.432 0.380 0.368 0.338

(25.7) (22.6) (21.9) (20.1)

0.378 0.322 0.296 0.275

D 0.153 0.166 0.185 0.175

(8.8) (9.6) (10.7) (10.1)

0.135 0.126 0.147 0.135

III E 0.291 0.224 0.217 0.219

(15.0) (11.5) (11.2) (11.3)

0.294 0.224 0.166 0.202

F 0.376 0.374 0.397 0.360

(10.9) (10.8) (11.5) (10.4)

0.493 0.537 0.320 0.507

I�II�III A�/F 0.281 0.247 0.235 0.2370.325 0.338 0.318 0.343

Pre-classified

I A 0.055 0.037 0.038 0.031

(7.6) (5.1) (5.3) (4.3)

0.047 0.034 0.037 0.034

B 0.068 0.042 0.048 0.039

(9.4) (5.8) (6.7) (5.4)

0.061 0.038 0.050 0.035

II C 0.113 0.073 0.088 0.048

(6.4) (4.1) (5.0) (2.7)

0.132 0.082 0.098 0.051

D 0.043 0.038 0.046 0.020

(2.5) (2.2) (2.7) (1.2)

0.062 0.048 0.046 0.022

III E 0.113 0.097 0.083 0.097

(5.6) (4.8) (4.1) (4.8)

0.168 0.200 0.120 0.143

F 0.055 0.066 0.091 0.050

(1.6) (1.9) (2.6) (1.4)

0.064 0.069 0.132 0.041

I�II�III A�/F 0.232 0.078 0.095 0.0810.286 0.166 0.167 0.162


by approximately a factor of three, i.e. to the average

levels in Table 5, reinforces the above arguments but

also indicates that this family of LDPE was produced

under distinct non-linear process dynamics.

To illustrate the genesis of the average errors given in

Table 5, i.e. the detailed performance of the sensors built

with sequential training, Fig. 5 shows the measured and

predicted time-records for grade I(A). The linear model

is unable to follow the time-sequence of the measured

MI, while clustering average sensor and particularly the

DynaRBF model perform reasonably well, consistently

with the average errors in Table 5. The performance of

fuzzy ARTMAP, while being reasonable on the average

(see Table 5), yields a step-like response around the

average MI. This is again a clear indication of the

insufficient training information provided by the se-

quential set. The fact that this effect is more evident for

the usually highly performing fuzzy ARTMAP algo-

rithm in difficult classification problems arises from its

need for increased training for periodic signals, as

illustrated in the benchmark reported by Carpenter et

al. (1992) and in tests carried out during the course of

the current study. The plant data for all grades have an

underlying periodicity around 1.1 residence time t units.

The performance of the single grade models devel-

oped with all variables and trained with the pre-

classified set of MI values is summarized in the lower

part of Table 5. The average performance of all virtualsensors for single grades trained using this pattern pre-

classification selection procedure is significantly better

in terms of both mean errors and standard deviations

for all product grades than that reported in the upper

part of Table 5 for sequential training. This effect is

most significant for the grades with less number of

patterns in Table 2, i.e. grades II(C), II(D) and III(E),

where errors in the predictions decrease by more than afactor of three with the change of training sets. It is also

more noticeable in the fuzzy ARTMAP sensor for its

especial sensitivity to training in systems with periodical

behavior, as explained before. For instance, the absolute

mean error of predictions obtained for grade II(C) with

the DynaRBF model drops from 0.456 (27.1%) to 0.149

(8.6%) and from 0.460 (27.4%) to 0.131 (7.6%) for fuzzy

ARTMAP. The same applies to grade III(E) and for theother grades with errors decreasing between 30 and 60%.

The better performance of the single grade neural

sensors with respect to the linear model is more evident

and consistent when training is carried out with the pre-

classified sets of data. The average relative errors for the

former models range from 2.8 to 8.6% while for the

latter they range from 6.1 to 15.1%. This tendency is

even clearer in terms of standard deviations. The averageperformance of DynaRBF and fuzzy ARTMAP are

similar, with an overall relative error of 5.2%, followed

by clustering average with 5.7%, and the linear model

with 10.6%. The classification capabilities of fuzzy

ARTMAP can be inferred by the lower standard

deviations of predictions that are obtained when the

more appropriate pre-classified set of data is used to

train the networks. The results for the three neuralsensors are remarkable considering the 9/2% errorassociated with the on-line MI measurements.

A detailed comparison of performances between the

linear model and the DynaRBF sensor is shown in Fig. 6

for the three product families. Under the training

scheme with pre-classified set of patterns detailed results

can only be evaluated in terms of deviations of predicted

MI with respect to measured values since trainingpatterns are not presented to the neural models accord-

ing to the time-sequence of plant measurements. Fig. 6

confirms the improved performance obtained with the

DynaRBF model during testing. This figure also shows

that there is an overlap in the MI distribution between

product grades II(C) and III(E). This could be one of the

reasons why product grade II(C) exhibits the maximum

errors in Table 5.The potential of the currently proposed neural sensing

technology should become more evident when attempt-

ing to predict the behavior of the ensemble of LDPE

grades simultaneously with only one sensor, i.e. with a

Fig. 5. Comparison of measured with predicted MI for grade I(A)

using single neural models with sequential training and all available

variables. (a) Linear; (b) clustering average; (c) fuzzy ARTMAP; (d)

DynaRBF.


composite sensor model. As a consequence, the four

models have also been tested with the more difficult

problem of forecasting the quality of the three LDPE

families (I�/II�/III) simultaneously, i.e. forecastinggrade transitions. Table 5 also summarizes the compo-

site model results obtained with both training sets

formed by the 14 203 patterns indicated in Table 2 and

using the 25 process variables listed in Table 1 com-

plemented by the normalized label to identify the six

grades. The composite models were tested with the

remaining 600 patterns, so that the test to train ratio was

kept comparable to that for single grade models.

Table 5 indicates that the absolute mean errors for the

three composite neural sensor models with sequential

training are approximately equal to 0.157 compared to

0.295 for the composite linear correlation model. This

expected good performance of the neural systems, also

reflected by the respective standard deviations of 0.153

and 0.301, is examined in Fig. 7 for a production cycle

including the three families. The behavior of the three

neural sensors is similar over time according to the time

sequence of the plots of the errors (Fig. 7b�/d) withrespect to the measured MI (Fig. 7a). Predictions made

over patterns corresponding to family I (grades A and

B) show the lowest fluctuations in error while those forfamily III are the highest, as was also the case for the

single grade sensors in sequential training in Table 5.

This behavior is partially due to deficient training as

discussed previously. Note that the MI variation shown

in Fig. 7a also illustrates the differences in process

dynamics between product families. The use of the most

suitable training set obtained by pre-classification in the

four composite models reduces absolute errors andstandard deviations of predictions from 0.157 and

0.153 to 0.121 and 0.116, respectively, for all virtual

sensor models and from 0.295 and 0.301 to 0.182 and

0.178 for the linear correlation model, as shown also in

Table 5. In this case of composite models the perfor-

mance of neural sensors is also better.

6.2. Models using the reduced set of variables

All single and the composite sensor models developed

for the reduced set of process variables selected by

dissimilarity of SOMs have been trained and tested by

Fig. 6. Comparison of measured MI time-records with predictions

obtained using the (a) single grade linear model and (b) single grade

DynaRBF model trained with the pre-classified dataset of data for the

three families of LDPE grades studied.

Fig. 7. Time-records of measured MI and for the errors of predictions

obtained with composite neural sensor models applicable simulta-

neously to all the families of LDPE considered and trained with the

sequential set of data. (a) measured MI; (b) clustering average; (c)

fuzzy ARTMAP; (d) DynaRBF.


using the datasets given in the lower part of Table 2. The

reduction in the number of input variables has the

advantages of (i) dealing with a lower dimensional

problem, (ii) cutting-back any noise that could contam-

inate the measurements of the discarded variables, and

(iii) avoiding variables that could provide conflicting

information with respect to more correlated variables in

relation to the target MI. Fig. 8 shows the variation of

the average cumulative dissimilarity between SOMs that

has been calculated for each grade by successive

insertion of the variables listed in Table 4, according

to Appendix A. The variables located beyond the

minimum dissimilarity point in the plots of Fig. 8 or

below the separation line in Table 4 do not contribute

with any additional relevant information to explain the

qualitative and quantitative behavior of MI and can be

discarded from the input process plant information.

It should be noted that the same variables but in a

different order would have been selected if an ordering

criteria based on both topological and correlation

information (Espinosa et al., 2001b) would have been

adopted. The application of this selection procedure

allowed an approximate 50% reduction in the number of

input data needed to develop all single sensor models.

Table 6 summarizes the performance of all single grademodels using the reduced set of input variables and

training with sequential and pre-classified data.

The effects of variable reduction when the single

grade sensor models were trained with the set of patterns

selected sequentially are summarized in the upper part

of Table 6. Comparison with the corresponding results

in Table 5 shows that the performance of all single grade

models, linear and neural, is maintained or improvesslightly despite the reduction in the information pro-

vided to the virtual sensor. For example, the single

models for grade I(B), all built with the first 15 variables

listed in Table 4 according to the minimum dissimilarity

in Fig. 8, yield predictions with an average absolute

error and standard deviation of 0.062 and 0.053,

respectively, for sequential training in Table 6, which

is slightly better than the corresponding 0.064 and 0.059for all 25 variables in Table 5. The same holds for the

more difficult to predict grade II(C), where the 26.9%

average relative error obtained for all variables with

sequential sets (Table 5) drops to about 21.5% (Table 6)

for the reduced set of 13 input variable selected from

Fig. 8 and Table 4. In the case of grade III(E) the

reduction in relative error is from 15.2% for all variables

to 11.3% using 12 variables. Similar behaviors areobserved for the other grades, both in terms of mean

errors and standard deviations.

The effects of variable reduction when the single

grade sensor models were trained with the best set of

patterns selected by the pre-classification procedure can

be observed in the lower part of Table 6. The errors of

predictions drop on the average from approximately 5%

in Table 5 to 4% in Table 6 for the neural sensors, andfrom 10 to 5.5% for the linear model. The reduction in

input variables also causes a decrease in standard

deviations. The same tendency of improvement or

comparable performance is observed for each individual

sensor and grade. DynaRBF is confirmed in Table 6 as

the best single grade neural sensor with an overall

absolute error and standard deviation of 0.048 (3.3%)

and 0.054 for the six grades A�/F, followed by clusteringaverage with 0.059 (4%) and 0.079, and fuzzy ARTMAP

with 0.066 (4.4%) and 0.081. In all cases, the mean errors

of predictions are comparable to the 9/2% errorassociated with MI on-line measurements. As discussed

before, the unexpected poorer performance of the

powerful ARTMAP classifier is due to insufficient

training considering the underlying periodicity of the

inferred target MI variable. The analyses of detailedinput pattern classification and MI forecasting during

testing support the average results in Table 6, but have

not been included here for brevity. Comparison between

the upper and lower parts of Table 6 confirms again the

Fig. 8. Average dissimilarity between SOMs for the selection of a

reduced and representative set of input variables for each and for the

ensemble of product grades.


adequacy of the pre-classification approach to select

data for training all sensors.

This significant improvement in the performance of

properly trained single grade virtual sensors caused bythe adequate reduction in input information indicates

that redundancy of information may be disadvanta-

geous when dealing with field variables contaminated by

measurement errors. Also, the inclusion of variables

with conflicting or contradicting information in relation

to that contributed by other variables with higher

correlations with the target MI could result in a

detrimental effect. Note that beyond the minimumpoints in the plots of Fig. 8 dissimilarity changes very

slowly with the addition of more variables indicating

that their inclusion or exclusion will not affect informa-

tion but could contribute to the addition or subtraction

of noise and/or of conflicting information with respect

to their effect on MI.

The effects of the reduction of variables in the

composite models are also included in Table 6 and themost relevant results highlighted in Fig. 9. The arrow in

Fig. 9a indicates that the minimum dissimilarity is

reached in this case when the first 17 variables are

considered from the ordered list in the last column of

Table 4. It is also clear from this figure that noise and

experimental errors make the choice of the minimum

dissimilarity more difficult. All composite models have

been developed with an input formed by these 17variables plus the normalized product grade label to

facilitate product identification.

The performance of the linear composite model

improves slightly with variable reduction when sequen-

tially trained, with absolute errors decreasing from 0.295

in Table 5 to 0.281 in Table 6, but worsens when the best

pre-classification set of data is used at the learning stage,

increasing from 0.182 in Table 5 to 0.232 in Table 6. Itshould be noted that the simple additive nature of this

model (Eq. (15)) could more easily cancel the noise

effects and/or handle conflicting information and, thus,

gain from the redundant information brought by any

extra variable when appropriately trained; the lowest

error of 0.182 and standard deviation of 0.178 corre-

spond to the composite linear correlation built with all

variables and trained with the pre-classified set ofpatterns (Table 5).

The functioning of the composite neural sensor

models trained sequentially worsens significantly with

the reduction of input variables; relative errors increase

on the overall from 9% in Table 5 to 14% in Table 6,

while standard deviations approximately double. For

the more appropriate training set selected by pre-

classification performance remains unchanged or im-proves slightly with variable reduction in terms of errors

but the dispersion of predictions increase. These results

indicate the difficulties encountered by the sensors to

properly classify certain input patterns during testing

and to generate adequate outputs for MI for the large

variety of process dynamics that occur in the LDPE

plant. The best composite neural sensors are dynaRBF

and clustering average, which yield the lowest absolute

errors of about 0.080 for the case of pre-classified

training (lower part of Table 6). The corresponding

error for fuzzy ARTMAP is an acceptable 0.095

considering the periodicity effects discussed before.

When the inadequate sequential training is applied in

this case of less input information (reduced variables)

the mean absolute error of predictions triples (upper

part of Table 6), which is the highest in Tables 5 and 6

for neural composite models. This is consistent with the

fact that any model deficiency should become more

evident when dealing with the more difficult problem of

predicting simultaneously all grades with reduced input

information. These results are illustrated and corrobo-

rated in detail in Fig. 9b and c where the measured and

predicted MI values corresponding to the test set of

patterns are compared for the four composite models

trained with pre-classified data. The superior perfor-

mance of the neural models compared to the linear

model is clear in these figures. The origin of the observed

Fig. 9. Results obtained for the composite models with all sensors

trained using the best set of pre-classified data and the reduced set of

input variables. (a) Variation of average dissimilarity between SOMs

with the inclusion of ordered variables; (b) measured vs. predicted MI

for the linear and clustering average models; (c) measured vs. predicted

MI for the fuzzy ARTMAP and DynaRBF models.


deviations is similar to that illustrated in Fig. 7 for the

complete set of variables and sequential training. Again,

the lowest deviations correspond to family I.

7. Concluding remarks

A neural network-based methodology to design and

build virtual sensors to infer product quality from

process variables has been developed and tested. Three

neural systems, together with a linear model, have been

used to build different virtual sensor models. A pre-dictive fuzzy ARTMAP algorithm is one of the archi-

tectures considered. The other two architectures,

identified as clustering average and DynaRBF, have

been built by the combination of a dynamic unsuper-

vised clustering layer with two different supervised

mapping procedures to implement the desired outputs.

This hybrid approach facilitates the use of more

elaborate learning algorithms for the supervised layerwithout affecting the underlying infrastructure based on

the dynamic unsupervised clustering.

Self organizing maps (SOM) have been introduced

and proven effective to estimate the relevance of certain

features and patterns by means of dissimilarity measures

and to select from this quantitative information the

minimum number of process variables needed as input

to the sensors. The neural sensors have been trainedusing data selected either by sequential order from the

time records of variables or by pre-classification of plant

data as a function of the difficulty to classify events

within the pool of available plant information.

As a proof-of-concept of the generic virtual sensor

model, three types of neural models have been devel-

oped to infer the MI of LDPE so that the accuracy of

on-line correlation based techniques that are commonlyused in industry is increased. Both single sensors for

each LDPE grade and a composite model for all grades

simultaneously have been implemented and tested. The

three neural models for single grades, with all process

variables measured at the beginning of the production

cycle considered as input, predict MI with relative mean

errors and standard deviations of approximately 5%

when appropriately trained with pre-classified patterns,compared with the average errors and standard devia-

tions of approximately 10 and 15%, respectively, ob-

tained for the corresponding linear correlation models.

A reduction in the number of variables up to 50% by

dissimilarity measures of SOM decreases these errors for

single grade neural and linear models to approximately

4 and 5.5%, respectively, with comparable decreases in

standard deviations. This reduction, which sets theaccuracy standards of virtual sensors close to the 9/2%experimental error for on-line MI measurements, could

be explained in terms of noise reduction and elimination

of conflicting input information with respect to the

target variable MI.

DynaRBF yields slightly more accurate predictions of

MI than fuzzy ARTMAP and clustering average forsingle grade sensors. It is well known that fuzzy

ARTMAP needs an extra amount of training informa-

tion when the problem under study possesses some

underlying periodicity, as is the case of the current on-

line time-variation of MI. The results obtained both for

single grade and composite models indicate that a neural

implementation provides prediction reliability and ac-

curacy. The proposed virtual sensors are capable oflearning the relationships between process variables

measured at the beginning of the production cycle and

the quality of the final product. Their superior perfor-

mance does not require any readjustment of parameters

during production cycles.

The three neural sensors perform similarly for com-

posite models. Sensors built with the reduced set of

input process variables and trained by pre-classificationyield the best predictions. The out-performance of

neural sensors with respect to linear correlations is

even more evident in this case of composite models

since neural sensors are capable of quickly adapting to

new operating conditions of the plant, including grade

transitions. Nevertheless, the effect of the reduction of

input variables increases the standard deviation indicat-

ing that better training is needed.

Acknowledgements

This research was supported by DGICYT (Spain),

projects PB96-1011 and PPQ2000-1339, and Comissio-

nat per a Universitats i Recerca (Catalunya), projects

1998SGR00102 and 2000SGR00103.

Appendix A: The Self Organizing Map

The SOM algorithm (Kohonen, 1990) performs a

topology-preserving mapping from a high-dimensional

input space onto a low dimensional output space formed

by a regular grid of map units. The lattice of the grid canbe either rectangular or hexagonal. This neural model is

biologically plausible (Kohonen, 1993) and is present in

various brains’ structures. From a functional point-of-

view, the SOM resembles vector quantization (VQ)

algorithms. These algorithms approximate, in an un-

supervised way, the probability density functions of

vectors by finite sets of codebook or reference vectors.

The only purpose of these methods is to describe classborders using a nearest-neighbor rule (see for example

the K -means algorithm in MacQueen (1967)). In con-

trast, SOM’s units are organized over the space spanned


by the regular grid and the whole neighborhood is

adapted in addition to the winner unit or neuron.

Each unit of the map is represented by a weight

vector, mi� [mi1; . . . ; min]T ; where n is equal to the

dimension of the input space. As in VQ, every weight

vector describing a class is called a codebook. Each unit

has a topological neighborhood Ni determined by the

form of the grid’s lattice, either rectangular or hexago-

nal. The number of units, as well as their topological

relations, is defined (and fixed) at the beginning of the

training process. The granularity (size) of the map will

determine its subsequent accuracy and generalizationcapabilities. During its training process, the SOM forms

an elastic net that folds onto the cloud formed by the

input data, trying to approximate the probability

density function of the original data by placing more

codebook vectors where the data are dense and fewer

units where they are sparse.

Training of the SOM proceeds as follows. At each

training step, one sample pattern x is randomly chosenfrom the training data. Similarities (distances) between x

and the codebook vectors are computed (the Euclidean

distance is usually adopted), looking for the BMU or

neuron. This similarity matching can be expressed as:

jx�mbmuj�minj xi �mij

(A1)

After finding the BMU and its topological neighbor

cells, their degree of matching is increased by moving

their codebook vectors in the proper direction in theinput space. This competitive learning process follows a

winner-takes-all approach, which can be described by

the following rules:

mi(t�1)�mi(t)�a(t)[x(t)�mi(t)]; i � Nbmu(t)mi(t); iQNbmu(t)

(A2)

In this equation t denotes time (position), Nbmu(t) is a

decreasing neighborhood function around the best

matching unit (BMU), and a (t) is a monotonically

decreasing learning rate. Two modes of operation for

the SOM can be considered in relation to the variationof the learning rate during training. An initial ordering

phase in which the map is formed and tuning is coarse,

and a later convergence phase, where the fine-tuning of

the codebook vectors is performed. This adaptive

approach corresponds to Hebbian learning.

A.1. Measures of dissimilarity for Kohonen maps

To select the most relevant input features the similar-

ity between the maps for each variable in relation to MI

must be measured. Several measures have been pro-posed to compare the positions of the reference vectors

in different map structures (Bauer and Pawelzik, 1992;

Kiviluoto, 1996; Willmann et al., 1994; Kraaijveld et al.,

1992). In the present work the dissimilarity measure

proposed by Kaski and Lagus (1997) is used. This

measure is based on a goodness measure proposed by

the same authors that combines an index of thecontinuity of the mapping from the dataset to the map

grid with a measure of accuracy of the map. The

dissimilarity between two maps L and M is defined by

the average difference of their goodness :

D(L; M)�E�jdL(x) � dM(x)j

dL(x) � dM(x)

�(A3)

In this equation E is the average expectation, and d (x )

the distance from x to the second BMU, denoted by

mbmu?(x ), beginning at the first BMU or winner neuron,

denoted by mbmu(x ). Of all possible paths between

mbmu(x ) and mbmu?(x ) the shortest path passing continu-

ously between neighbor units is selected,

d(x)�kx�mbmu(x)k�mini

XKbmu?(x)�1k�0

kmIi(k)

�mIi(k�1)k (A4)

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